# Totally nonparallel immersions

## Geometry Luncheon Seminar

## Meeting Details

For more information about this meeting, contact Kristin Berrigan, Dmitri Burago.

**Speaker:** Michael Harrison, Carnegie-Mellon U.

**Abstract:** An immersion from a smooth n-dimensional manifold M into R^q is called totally nonparallel if, for every distinct points x and y in M, the tangent spaces at f(x) and f(y) contain no parallel lines. The simplest example is the map R \to R^2 sending x to (x,x^2). Given a manifold M, what is the minimum dimension q = q(M) such that M admits a totally nonparallel immersion into R^q? I will discuss one method of studying this problem using the h-principle, a powerful tool in differential topology which is used to study spaces of functions with certain distinguished properties (immersions, submersions, embeddings, free maps, etc.). I will give a neat proof of a special case of the Smale-Hirsch Theorem, using the "removal of singularities" h-principle technique due to Eliashberg and Gromov. I will discuss how to adapt this argument to study totally nonparallel immersions.

## Room Reservation Information

**Room Number:** 114 McAllister

**Date:** 10/09/2019

**Time:** 12:00pm - 1:30pm